Question

Given z^2=1/2(cos(2pi/5)+i sin(2pi/5)), which lettered point represents z?

Answer 1

Answer:

$z$ is represented by point A.

Step-by-step explanation:

According to the De Moivre's Formula, the power of a complex number of the form $z = r\cdot (\cos \theta + i\,\sin \theta)$ is defined by this formula:

$z^{n} = r^{n}\cdot [\cos n\theta + i\,\sin n\theta]$ (1)

Where:

$r$ - Norm of the complex number, dimensionless.

$\theta$ - Direction of the complex number, measured in radians.

$n$ - Power of the resulting complex number, dimensionless.

Given that $z^{2} = \frac{1}{2}\cdot \left[\cos \left(\frac{2\pi}{5} \right)+ i\,\sin \left(\frac{2\pi}{5} \right)\right]$, then the following variables are:

$r^{2} = \frac{1}{2}$ (2)

$2\theta = \frac{2\pi}{5}$ (3)

And the norm and direction of the complex number are, respectively:

$r = \frac{\sqrt{2}}{2}$ and $\theta = \frac{\pi}{5}$

Then, the complex number is:

$z = \frac{\sqrt{2}}{2}\cdot \left[\cos \left(\frac{\pi}{5} \right)+i\,\sin \left(\frac{\pi}{5} \right)\right]$

$z = 0.572 + i\,0.416$

Which corresponds to point A.